Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

-e time-fractional order equation [1–4] has wide applications in viscoelastic materials, signal processing, fluid mechanics, and dynamic of viscoelastic materials. -e analytic solution to this equation is almost impossible to obtain. In recent years, several numerical methods [5–11] have been proposed. In previous work, the author used Taylor’s formula or Delta function to construct reproducing kernel space [12–17]. In this paper, we structure some new reproductive kernel spaces based on Jacobi polynomials and give a numerical solution of a class of time-space fractional order diffusion equation using piecewise reproducing kernel method.


Introduction
In this paper, we consider the following time-fractional order diffusion equation: where β 1 (x, t), β 2 (x, t), f(x, t) are known functions, and D c t u(x, t) is the variable order Caputo fractional derivative of order c: e time-fractional order equation [1][2][3][4] has wide applications in viscoelastic materials, signal processing, fluid mechanics, and dynamic of viscoelastic materials. e analytic solution to this equation is almost impossible to obtain. In recent years, several numerical methods [5][6][7][8][9][10][11] have been proposed. In previous work, the author used Taylor's formula or Delta function to construct reproducing kernel space [12][13][14][15][16][17]. In this paper, we structure some new reproductive kernel spaces based on Jacobi polynomials and give a numerical solution of a class of time-space fractional order diffusion equation using piecewise reproducing kernel method.

Structing Reproductive Kernel Space
Based on the Shifted Jacobi Polynomials e shifted Jacobi polynomials P α,β 1,i (x) of degree i is given [18] by where e shifted Jacobi polynomials on the interval x ∈ [0, 1] are orthogonal with the orthogonality condition where ω(x) � x β (1 − x) α is a weight function and is the weighted inner product space of the shifted Jacobi polynomials on [0, 1]. e inner product and norm are defined as where P α,β From [17,19], we can prove that H n [0, 1] is a reproducing kernel Hilbert space. Its reproducing kernel is where . Using Ref. [5] and the reproducing kernel of H n [0, 1], we can get following reproducing kernel spaces.
the same inner product as H n and H n is a reproducing kernel space and its reproducing kernel is with the same inner product as H n+1 and H n+1 is a reproducing kernel space and its reproducing kernel is where K n+1 (x,y)�R n+1 (x,y)− (R n+1 (0,x)R n+1 (y,0)/ ‖R n+1 (0,0)‖ 2 ).
Reproducing kernels with different α, β are shown in

Piecewise Reproducing Kernel Method
After homogenization, equation (1) is converted to the following form: (16) where the β ik are the coefficients resulting from Gram-Schmidt orthonormalization.
Deriving from the form of (17), we get the approximate solution of (14) as However, the direct application of (17) could not have a good numerical simulation effect possibly for (1). e focus of this paper is to fill this gap, so we use the piecewise reproducing kernel method. e main technique of the piecewise reproducing kernel method see Ref. [16,21,24,25].

Numerical Experiments
In this section, some numerical experiments are studied to demonstrate the accuracy of the present method. Experiment 1. We consider the following time fractional reaction-diffusion equation: . Numerical solution of Experiment 1 is shown in Figures 9-11 and Table 1. From Table 1, we can see that the absolute error is getting smaller and smaller when h is smaller. Figure 10 shows the relationship between absolute error and reproducing kernel. Figure 11 shows the relationship between absolute error and c.

Experiment 2.
We consider the following time fractional reaction-diffusion equation [ . Numerical solution of Experiment 2 is shown in Table 2. From Table 2, we can see that the absolute  error obtained by the present method is smaller than the absolute error obtained by Ref. [7].

Experiment 4.
We consider the following time-space fractional advection-reaction-diffusion equation:  Table 4 by present method at n � 2, N � 5. Figures 13 and 14 show the relationship between absolute error and α, β. From Figures 15 and 16, we can see that the absolute error is small. Figures 17-20 show the relationship between absolute error and c.

Conclusions
In this paper, some new reproductive kernels are given. e numerical results of some models show that the present method has high precision compared with traditional RKM, and has a better convergence for this kind of model. Besides, the method can also be used to study other time variable fractional order advection-dispersion model.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.